1,5

As we showed in A160485 the nth term of the coefficients of matrix row BS1[1-2*m,n] for m = 1 ,2, 3, .. , can be generated with the RBS1(1-2*m,n) polynomials.

We define the o.g.f.s. of these polynomials by GFRBS1(z,1-2*m) = sum(RBS1(1-2*m,n)*z^(n-1), n=1..infinity) for m = 1, 2, 3, .. . The general expression of these o.g.f.s. is GFRBS1(z,1-2*m) = (-1)*RB(z,1-2*m)/(z-1)^m.

The RB(z,1-2*m) polynomials lead to a triangle that is a subtriangle of the 'double triangle' A008971. The even rows of the latter triangle are identical to the rows of our triangle.

The Maple program given below is derived from the one given in A008971.

The first few rows of the triangle are:

[1]

[1, 1]

[1, 18, 5]

[1, 179, 479, 61]

[1, 1636, 18270, 19028, 1385]

The first few RB(z,1-2*m) polynomials are:

RB(z,-1) = 1

RB(z,-3) = z+1

RB(z,-5) = z^2+18*z+5

RB(z,-7) = z^3+179*z^2+479*z+61

The first few GFRBS1(z,1-2*m) are:

GFRBS1(z,-1) = (-1)*(1)/(z-1)

GFRBS1(z,-3) = (-1)*(z+1)/(z-1)^2

GFRBS1(z,-5) = (-1)*(z^2+18*z+5)/(z-1)^3

GFRBS1(z,-7) = (-1)*(z^3+179*z^2+479*z+61)/(z-1)^4

nmax:=15; G:=sqrt(1-t)/(sqrt(1-t)*cosh(x*sqrt(1-t))-sinh(x*sqrt(1-t))): Gser:= simplify(series(G, x=0, nmax+1)): for m from 0 to nmax do P[m]:=sort(expand(m!* coeff(Gser, x, m))) od: nmx:=floor(nmax/2); for n from 0 to nmx do for k from 0 to nmx-1 do A(n+1, n+1-k):=coeff(P[2*n], t, n-k) od: od: for n from 1 to nmax do for m from 1 to n do b((n-1)*(n)/2+m):=A(n, m) od: od: a:=m-> b(m): seq(a(m), m=1..(nmx)*(nmx+1)/2);

Cf. A160480 and A160485.

The row sums equal A010050.

This triangle is a sub-triangle of A008971.

A000340(2*n-2), A000363(2*n+2) and A000507(2*n+4) equal the second, third and fourth left hand columns.

The first right hand column equals the Euler numbers A000364.

Sequence in context: A059654 A080694 A040314 this_sequence A040312 A065909 A038642

Adjacent sequences: A160483 A160484 A160485 this_sequence A160487 A160488 A160489

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009